where \(T_{\mathrm {sys}}\)  is the total system noise temperatureFor the GBT, T\(_{\mathrm {sys}}\)  is measured in the T\(_{\mathrm {a}}\)  scale, while for the IRAM 30-m telescope,
it is measured in the T\(_{\mathrm {a}}^*\)  scale, which includes atmospheric attenuation and forward beam efficiency corrections: T\(_{\mathrm {a}}^*\)  = T\(_{\mathrm {a}}\times \frac{e{^{tau}}}{\eta _{\mathrm {ffs}}}\) 
[1]}, including the receiver
noise and all noise from the sky and ground spillover, \(\delta \nu \)  is the spectral resolution, and \(\tau \)  the integration
time. Here, \(\eta \)  is sometimes introduced to take into account other losses such as one- or two-bit autocorrelator conversion losses. When using beam-switching or position-switching
modes, a second spectrum of similar characteristics is obtained to be subtracted from the first
one. Because the noise fluctuations of both spectra are uncorrelated, the subtraction increases the noise fluctuation temperature by \(\sqrt{2}\) .
If we consider that \(\tau \)  represents the total (ON+OFF) time, then the noise fluctuation temperature increases by another \(\sqrt{2}\) , as only half of the time
was spent on each individual spectrum, effectively doubling the
final noise fluctuation of the spectrum (and more time is lost in overheads, like the mechanical displacement of mirrors, or the rotation of the whole telescope).
If the frequency-switch mode has been used instead, then the OFF spectrum becomes a second ON
spectrum and the penalty is only \(\sqrt{2}\)   but at the expense of spoiling the baseline because of the frequency dependence
of the receiver (and the standing wave patterns) that prevents an exact
superposition of the baseline structure from the ON1 and ON2 spectra.
Improved strategies consist of integrating over longer periods of time for the OFF spectrum and
subtracting the same OFF spectrum from several ONs, minimizing the time spent OFF source
and the noise fluctuations added in the subtraction. These strategies require very stable receivers and sky conditions to be efficient, but can marginally beat
the performance of the frequency-switch mode in terms of noise fluctuations and provide much flatter baselines; however, they become less efficient in the end if spatial smoothing is applied, because the OFF is identical for the adjacent ON-OFF pairs such
that their average does not diminish the noise fluctuations efficiently,
and then frequency-switching becomes preferable.
